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Chapter 2: Problem 23

If \(V, W\) are two varieties over an algebraically closed field \(k,\) and if \(V \times W\) is their product, as defined in (I, Ex. 3.15,3.16 ), and if \(t\) is the functor of (2.6) then \(t(V \times W)=t(V) \times_{\text {spec } k} t(W)\).

Short Answer

Expert verified
To prove that \(t(V \times W) = t(V) \times_{\text {spec } k} t(W)\), apply the functor \(t\) to \(V \times W\) and the product of \(t(V)\) and \(t(W)\) over \(\text{spec } k\), and then demonstrate that these two presheafs are equal by showing a natural bijection between their sets of morphisms from an arbitrary scheme \(X\).

Step by step solution

01

Understanding the Objective

First, understand the task at hand. We are given that \(V, W\) are two varieties over an algebraically closed field \(k\), and \(t\) is a functor. The task is to show that \(t(V \times W) = t(V) \times_{\text {spec } k} t(W)\).
02

Applying Functor \(t\)

Apply the functor \(t\) to the direct product \(V \times W\). This transforms the product of varieties into a presheaf over the category of affine schemes.
03

Applying Product of Functors

Next, construct the product of the transformed varieties \(t(V)\) and \(t(W)\) over \(\text{spec } k\). This yields a presheaf over the category of affine schemes.
04

Showing Equality of Presheafs

Now, show that the set of morphisms from an arbitrary scheme \(X\) into the presheaf \(t(V) \times_{\text {spec } k} t(W)\) is in natural bijection with the set of morphisms from \(X\) into the presheaf \(t(V \times W)\). This demonstrates the required equality of the presheaves, thus proving the original statement.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Varieties
In algebraic geometry, varieties are fundamental objects of study. A variety can be thought of as the set of solutions to a system of polynomial equations. These equations are defined over a field, and in many cases, we work with varieties over an algebraically closed field like the complex numbers.
  • Algebraic varieties are important because they generalize the familiar notion of curves and surfaces to higher dimensions.
  • For example, a curve is a variety of dimension 1, while a surface has dimension 2.
  • Varieties come with a rich structure because they allow polynomial equations, which provide tools to analyze their geometry and properties.
Understanding varieties involves exploring their dimensions, singularities, and their relationships with other varieties, often through morphisms. Morphisms between varieties are map-like structures that preserve the algebraic operations.
Functor
Functors are mappings between categories that preserve the structures of those categories. In the context of algebraic geometry, a functor often maps between the category of varieties and some other category like sets or schemes. This concept helps to shift perspectives to better understand complex relationships.
  • A functor has two main properties: it associates every object in one category to an object in another, and it does similarly for morphisms between objects.
  • For example, if there is a morphism between two varieties, a functor would map this morphism to a corresponding morphism in the other category.
  • Preservation of composition and identities is key. In simpler terms, the functor respects the way objects interact and relate.
By using a functor, algebraic geometers can translate problems into a different setting where they might be simpler to study.
Algebraically Closed Field
An algebraically closed field is one in which every non-constant polynomial equation has a solution. A familiar example of an algebraically closed field is the field of complex numbers, \( \mathbb{C} \). Such fields have unique properties that are extremely useful in algebraic geometry.
  • One important property is that every geometric object defined over an algebraically closed field is more manageable since all necessary solutions already exist within the field.
  • When a field is algebraically closed, it simplifies many questions about varieties because any polynomial equation that we want to solve can do so within the field itself.
  • This makes algebraic geometry over such fields a nice blend of geometry and algebra, allowing for the deep and meaningful analysis of varieties.
Algebraically closed fields allow algebraic geometer's to understand the full scope of polynomial relationships without worrying about missing solutions.
Product of Varieties
The product of varieties is an advanced concept in algebraic geometry that combines two varieties into a new one. This is somewhat analogous to Cartesian products in set theory, but involves more complex algebraic structures.
  • Given two varieties, \( V \) and \( W \), their product \( V \times W \) is constructed by pairing their points in a consistent way that respects the algebraic structure of each variety.
  • This operation allows algebraic geometers to explore interactions between varieties and can lead to discovering new varieties.
  • The product construction is crucial in proving properties about morphisms, and in understanding how varieties interact over a given field.
The product of varieties is not just a theoretical construct; it has practical applications in areas like number theory and complex analysis, where understanding such relationships is key.

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Most popular questions from this chapter

(a) Let \(\varphi: \overline{\mathscr{H}} \rightarrow \mathscr{S}\) be a morphism of presheaves such that \(\varphi(\mathcal{U}): \mathscr{F}(U) \rightarrow \mathscr{G}(U)\) is injective for each \(U\). Show that the induced \(\operatorname{map} \varphi^{+}: \overline{\mathscr{H}}^{+} \rightarrow \mathscr{S}^{+}\) of associated sheaves is injective. (b) Use part (a) to show that if \(\varphi: \bar{y} \rightarrow \mathscr{G}\) is a morphism of sheaves, then im \(\varphi\) can be naturally identified with a subsheaf of \(\mathscr{G}\). as mentioned in the text.

Describe the spectrum of the zero ring, and show that it is an initial object for the category of schemes. (According to our conventions, all ring homomorphisms must take 1 to \(1 .\) since \(0=1\) in the zero ring, we see that each ring \(R\) admits a unique homomorphism to the zero ring, but that there is no homomorphism from the zero ring to \(R\) unless \(0=1\) in \(R\).)

Again let \(X\) be a noetherian scheme, and \(\mathscr{F}\) a coherent sheaf on \(X .\) We will consider the function \\[ \varphi(x)=\operatorname{dim}_{k(x)} \mathscr{F}_{x} \otimes_{\sigma_{x}} k(x) \\] where \(k(x)=O_{x} / m_{x}\) is the residue field at the point \(x .\) Use Nakayama's lemma to prove the following results. (a) The function \(\varphi\) is upper semi-continuous, i.e., for any \(n \in \mathbf{Z},\) the set \(\\{x \in X | \varphi(x) \geqslant n\\}\) is closed. (b) If \(\mathscr{F}\) is locally free, and \(X\) is connected, then \(\varphi\) is a constant function. (c) Conversely, if \(X\) is reduced, and \(\varphi\) is constant, then \(\mathscr{F}\) is locally free.

Product Schemes. (a) Let \(X\) and \(Y\) be schemes over another scheme \(S\). Use (8.10) and (8.11) to show that \(\Omega_{X \times_{s} Y / S} \cong p_{1}^{*} \Omega_{X / S} \oplus p_{2}^{*} \Omega_{Y / S}\) (b) If \(X\) and \(Y\) are nonsingular varieties over a field \(k\), show that \(\omega_{X \times Y} \cong p_{1}^{*} \omega_{X} \otimes\) \(p_{2}^{*} \omega_{Y}\) (c) Let \(Y\) be a nonsingular plane cubic curve, and let \(X\) be the surface \(Y \times Y\) Show that \(p_{g}(X)=1\) but \(p_{a}(X)=-1\) (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different.

Let \(f: X \rightarrow Y\) be a morphism of separated schemes of finite type over a noetherian scheme \(S\). Let \(Z\) be a closed subscheme of \(X\) which is proper over \(S\). Show that \(f(Z)\) is closed in \(Y,\) and that \(f(Z)\) with its image subscheme structure (Ex. \(3.11 d\) ) is proper over \(S .\) We refer to this result by saying that "the image of a proper scheme is proper." [Hint: Factor \(f\) into the graph morphism \(\Gamma_{f}: X \rightarrow X \times_{s} Y\) followed by the second projection \(\left.p_{2}, \text { and show that } \Gamma_{f} \text { is a closed immersion. }\right]\)
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